Kinematics
AP Physics 1 Topic 1 9:04 English narration · English + 中文 subtitles burned in
Chapters
Transcript
Two balls leave a table at the same moment.
两个球在同一瞬间离开桌面。
One is simply dropped. The other is fired sideways, hard.
一个只是自由落下，另一个被用力水平射出。
Which one hits the floor first?
哪一个先落地？
<slow>They land together.</slow> Every time.
它们同时落地，每一次都是。
The sideways motion changes nothing about the falling — and that one fact is the key to this entire unit.
水平方向的运动完全不影响下落—— 而这一个事实，正是理解整个单元的钥匙。
This is Unit One: kinematics.
这是第一单元：运动学。
How things move, described precisely — without yet asking what makes them move.
精确地描述物体如何运动——但暂时不问是什么让它们运动。
Signs, graphs, equations, and the two-motion trick you just saw.
符号、图像、公式，以及你刚才看到的「两个运动」技巧。
Let's begin.
让我们开始吧。
Everything in this unit is one of two kinds of quantity.
本单元的一切都属于两类量之一。
A scalar has magnitude — size — only: distance, speed, time, mass.
标量只有大小：距离、速率、时间、质量。
A vector has magnitude and direction: displacement, velocity, acceleration, force.
矢量既有大小又有方向：位移、速度、加速度、力。
The difference is not fussy — it changes the answer.
这个区别并不是吹毛求疵—— 它会改变答案。
Walk three metres east, then one metre back west.
向东走三米，再向西走回一米。
The distance you covered is four metres. Your displacement is only two metres east.
你走过的距离是四米， 而你的位移只有向东两米。
In one dimension, direction is just a plus or a minus sign along an axis you choose.
在一维情况下，方向就是你所选坐标轴上的一个正号或负号。
Choose that positive direction first, before anything else.
在做任何事之前，先把正方向定下来。
Three linked vectors describe motion along a line.
三个相互关联的矢量描述沿直线的运动。
Displacement is the change in position — where you ended up, measured from where you started.
位移是位置的变化—— 从起点量到终点。
Velocity is how fast the position changes: displacement divided by the time taken.
速度是位置变化的快慢：位移除以所用的时间。
Its size alone is the speed.
它的大小就是速率。
Acceleration is how fast the velocity changes: change in velocity divided by time.
加速度是速度变化的快慢：速度的变化除以时间。
And keep average apart from instantaneous.
还要把平均值和瞬时值分开。
Average velocity uses the whole trip. Instantaneous velocity is the value at a single moment.
平均速度用的是整段行程； 瞬时速度是某一个时刻的值。
They agree only when the velocity never changes.
只有当速度始终不变时，两者才相等。
Now the mistake that costs the most marks.
现在来看最容易丢分的错误。
When velocity and acceleration have the same sign, the object speeds up.
当速度和加速度符号相同时，物体加速。
When they have opposite signs, it slows down — that is deceleration.
当它们符号相反时，物体减速。
That is the whole rule.
规则就是这样。
Here is the trap: a negative acceleration does not mean slowing down.
陷阱在于： 负的加速度并不意味着减速。
A ball falling faster and faster has a negative velocity and a negative acceleration — same sign, so it is speeding up.
一个越落越快的球，速度为负，加速度也为负—— 符号相同，所以它在加速。
And a velocity of minus five metres per second is not slow. It is five metres per second, going the other way.
而负五米每秒的速度并不「慢」， 它就是五米每秒，只是方向相反。
The same motion can appear four ways — a description in words, a graph, a table or an equation — and you should be able to move between them.
当加速度恒定时——也只有这时——你可以用三个公式解决几乎所有问题。 要记住它们，也要记住它们的适用条件。
When the acceleration is constant — and only then — you get three equations that solve almost everything. Learn them, and learn their limit. To pick one, list the three quantities you already know, plus the one you want.
选公式的方法是： 列出你已知的三个量，加上你要求的那一个。
The right equation is the one containing exactly those four, so only one unknown is left.
正确的公式恰好含有这四个量， 于是只剩下一个未知数。
A car starts from rest and accelerates at two metres per second squared for six seconds.
一辆汽车从静止开始，以二米每二次方秒加速六秒。
Final velocity: zero plus two times six, so twelve metres per second.
末速度：零加二乘六，等于十二米每秒。
Distance: a half times two times six squared, which is thirty-six metres.
距离：二分之一乘二乘六的平方，等于三十六米。
Graphs carry the same information, and the exam loves them.
图像承载同样的信息，而考试非常喜欢考它们。
On a position-time graph, the slope is the velocity.
在位移—时间图上，斜率就是速度。
Steeper means faster; flat means at rest; a curve means the velocity is changing.
越陡越快；水平表示静止；曲线表示速度在变化。
On a velocity-time graph, the slope is the acceleration — and the area under the line is the displacement.
在速度—时间图上， 斜率是加速度——而线下的面积是位移。
That second one is worth repeating. Slope gives you the rate; area gives you the total.
第二点值得重复： 斜率给出变化率，面积给出总量。
And area below the time axis counts as negative, because the object moved the other way.
而时间轴以下的面积记为负， 因为物体朝相反方向运动。
Try one.
试一道题。
A cyclist speeds up evenly from rest to eight metres per second in four seconds, then holds that speed for six more.
一位骑车人从静止在四秒内均匀加速到八米每秒，然后以这个速度再行驶六秒。
How far did she travel?
她一共走了多远？
Pause here and work it out.
先暂停，自己算一算。
Split the area into shapes you know: a triangle, then a rectangle.
把面积分成你熟悉的形状： 先是三角形，再是矩形。
The triangle is a half times four times eight, which is sixteen.
三角形是二分之一乘四乘八，等于十六。
The rectangle is six times eight, which is forty-eight.
矩形是六乘八，等于四十八。
So sixteen plus forty-eight gives sixty-four metres.
所以十六加四十八，得到六十四米。
Drop the air resistance and you have free fall: every object near Earth falls with the same constant acceleration.
忽略空气阻力，地球附近的所有物体都以相同的恒定加速度下落。
Near Earth's surface, AP asks you to use ten metres per second squared, pointing down.
在地球表面附近，AP 要求你使用十米每二次方秒，方向向下。
Throw a ball straight up at fifteen metres per second.
把一个球以十五米每秒竖直上抛。
At the very top its velocity is zero — but the acceleration is still ten downward, the whole time. It never pauses.
在最高点它的速度为零—— 但加速度始终是向下的十，一刻也没有停。
Height reached: fifteen squared over two times ten, which is eleven point two five metres.
上升高度： 十五的平方除以二乘十，等于十一点二五米。
Time up: one point five seconds.
上升时间：一点五秒。
And the motion is symmetric, so the fall takes the same one point five seconds — three seconds in the air.
而运动是对称的，所以下落也用一点五秒——在空中一共三秒。
Every velocity is measured against something — a reference frame.
每一个速度都是相对某个参考系测量的。
Walk forward on a moving train and you have one velocity relative to the train, and a different one relative to the ground. Neither is wrong.
在行驶的火车上向前走， 你相对火车有一个速度，相对地面有另一个速度，两者都不是错的。
Two metres per second on a thirty-metre-per-second train is thirty-two relative to the ground — both readings are correct.
在三十米每秒的火车上以两米每秒行走，相对地面就是三十二米每秒——两种读数都对。
To get the velocity of A relative to B, subtract B's velocity from A's.
要求 A 相对 B 的速度，就用 A 的速度减去 B 的速度。
A boat points straight across a river at three metres per second, and the current carries it along at four.
一条船以三米每秒垂直指向对岸，水流以四米每秒把它带向下游。
Those are at right angles, so add them like a triangle: the square root of three squared plus four squared is five metres per second, pointing fifty-three degrees downstream from straight across.
两者互相垂直，所以像直角三角形一样相加：三的平方加四的平方再开平方， 得到五米每秒，方向偏向下游五十三度。
One more word: an inertial frame is one that is not accelerating.
还有一个词：惯性参考系是不加速的参考系。
Acceleration comes out the same in every inertial frame — but in a non-inertial one, an accelerating car or a spinning roundabout, it does not.
在所有惯性系中测得的加速度都相同—— 但在非惯性系里，比如加速的汽车或旋转的转盘，就不再相同了。
In two dimensions there is one move, and you make it every single time. Break every vector into two perpendicular pieces — its components.
在二维情况下只有一个动作，而你每一次都要做它： 把每个矢量分解成两个互相垂直的部分——它的分量。
The horizontal part is the size times the cosine of the angle. The vertical part is the size times the sine.
水平分量是大小乘以角的余弦， 竖直分量是大小乘以角的正弦。
Then handle each axis completely separately, as its own one-dimensional problem, and recombine at the end.
然后把每个轴完全分开处理， 各自当作一维问题，最后再合起来。
Two dimensions never require new physics. They require the same physics, done twice.
二维从不需要新的物理， 它需要的是同样的物理，做两遍。
Now back to the two balls.
回到那两个球。
For anything moving under gravity alone, the two motions are completely independent.
对于只受重力作用的物体，两个方向的运动完全独立。
Horizontally there is no force, so the horizontal velocity never changes at all.
水平方向没有力，所以水平速度完全不变。
Vertically gravity pulls down with that constant ten.
竖直方向重力以恒定的十向下拉。
They share exactly one thing: the clock.
它们只共用一样东西：时间。
The same time runs in both.
同一个时间在两个方向上流逝。
That is why the trajectory — the path — is a parabola, and why the dropped ball and the fired ball land together — their vertical stories are identical.
这就是轨迹为抛物线的原因，也是那个自由落下的球和那个被射出的球同时落地的原因—— 它们在竖直方向上的经历完全相同。
Last one, and it is the classic.
最后一题，也是最经典的一题。
A ball is kicked at twenty metres per second, thirty degrees above the horizontal.
一个球以二十米每秒、与水平成三十度被踢出。
Find the time of flight, the greatest height, and the range.
求飞行时间、最大高度和射程。 先暂停，自己试一试。
Pause here and try it. Components first: horizontal is twenty times cosine thirty, about seventeen point three. Vertical is twenty times sine thirty, exactly ten.
先求分量： 水平分量是二十乘三十度的余弦，约为十七点三；竖直分量是二十乘三十度的正弦， 正好是十。
The vertical part sets the time: it takes one second to reach the top, so two seconds in the air.
竖直方向决定时间：上升到最高点用一秒，所以空中共两秒。
Greatest height is ten squared over twenty, five metres.
最大高度是十的平方除以二十，等于五米。
And the range is seventeen point three times two — about thirty-five metres.
射程是十七点三乘二——约三十五米。
Three habits that save marks.
三个能保住分数的习惯。
First, fix a positive direction before you write anything, and let every sign follow from it.
第一，在动笔之前先定下正方向，让所有符号都由它决定。
Second, the kinematic equations work only while the acceleration is constant — check that before you use one.
第二，运动学公式只在加速度恒定时成立——用之前先确认这一点。
Third, at the top of a projectile's flight the vertical velocity is zero, but the object is not at rest: the horizontal velocity is exactly what it always was.
第三，在抛体飞行的最高点，竖直速度为零，但物体并没有静止： 水平速度和一直以来完全一样。
Get those three, and Unit One is yours.
做到这三点，第一单元就是你的了。
Five leftover facts.
还剩五件事实。
First, kinematics describes how objects move, without asking why — and the same motion can be a description, a graph, a table, or an equation.
第一，运动学描述物体如何运动，不问为什么—— 同一运动可以是文字、图像、表格或公式。
Second, instantaneous velocity is the slope of the position-time graph at one point.
第二，瞬时速度是位移—时间图上某一点的斜率。
Third, there is a fourth kinematic equation: displacement equals half of initial plus final velocity, times time — still only while acceleration is constant.
第三，还有第四个运动学公式：位移等于初速度加末速度的一半，再乘时间—— 仍然只在加速度恒定时成立。
Fourth, at the top of a projectile the speed equals the unchanged horizontal velocity, not zero.
第四，抛体最高点的速率等于不变的水平速度，不是零。
Fifth, distinguish distance from displacement, and speed from velocity; a negative velocity means the other way, not slow.
第五，要分清距离与位移、速率与速度；负速度表示朝另一边，并不表示慢。